LMFDB - Embedded newform 675.5.c.h.26.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,5,Mod(26,675)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("675.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	675.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$69.7747250816$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.26
Dual form			675.5.c.h.26.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{2} +7.00000 q^{4} +19.0000 q^{7} +69.0000i q^{8} +123.000i q^{11} -302.000 q^{13} +57.0000i q^{14} -95.0000 q^{16} -414.000i q^{17} -304.000 q^{19} -369.000 q^{22} -300.000i q^{23} -906.000i q^{26} +133.000 q^{28} -678.000i q^{29} +239.000 q^{31} +819.000i q^{32} +1242.00 q^{34} -740.000 q^{37} -912.000i q^{38} +228.000i q^{41} +982.000 q^{43} +861.000i q^{44} +900.000 q^{46} -2166.00i q^{47} -2040.00 q^{49} -2114.00 q^{52} +1593.00i q^{53} +1311.00i q^{56} +2034.00 q^{58} -2922.00i q^{59} -316.000 q^{61} +717.000i q^{62} -3977.00 q^{64} -4622.00 q^{67} -2898.00i q^{68} +1818.00i q^{71} +3031.00 q^{73} -2220.00i q^{74} -2128.00 q^{76} +2337.00i q^{77} -10450.0 q^{79} -684.000 q^{82} -12633.0i q^{83} +2946.00i q^{86} -8487.00 q^{88} -7002.00i q^{89} -5738.00 q^{91} -2100.00i q^{92} +6498.00 q^{94} +6517.00 q^{97} -6120.00i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} + 38 q^{7} - 604 q^{13} - 190 q^{16} - 608 q^{19} - 738 q^{22} + 266 q^{28} + 478 q^{31} + 2484 q^{34} - 1480 q^{37} + 1964 q^{43} + 1800 q^{46} - 4080 q^{49} - 4228 q^{52} + 4068 q^{58}+ \cdots + 13034 q^{97}+O(q^{100}) $ 2 * q + 14 * q^4 + 38 * q^7 - 604 * q^13 - 190 * q^16 - 608 * q^19 - 738 * q^22 + 266 * q^28 + 478 * q^31 + 2484 * q^34 - 1480 * q^37 + 1964 * q^43 + 1800 * q^46 - 4080 * q^49 - 4228 * q^52 + 4068 * q^58 - 632 * q^61 - 7954 * q^64 - 9244 * q^67 + 6062 * q^73 - 4256 * q^76 - 20900 * q^79 - 1368 * q^82 - 16974 * q^88 - 11476 * q^91 + 12996 * q^94 + 13034 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.00000i	0.750000i	0.927025	+	0.375000i	$0.122357\pi$
$2$	3.00000i	0.750000i	−0.927025	+	0.375000i	$0.877643\pi$
$3$	0	0
$3$	0	0
$4$	7.00000	0.437500
$5$	0	0
$5$	0	0
$6$	0	0
$7$	19.0000	0.387755	0.193878	−	0.981026i	$-0.437894\pi$
$7$	19.0000	0.387755	0.193878	+	0.981026i	$0.437894\pi$
$8$	69.0000i	1.07812i
$9$	0	0
$10$	0	0
$11$	123.000i	1.01653i	0.861201	+	0.508264i	$0.169713\pi$
$11$	123.000i	1.01653i	−0.861201	+	0.508264i	$0.830287\pi$
$12$	0	0
$13$	−302.000	−1.78698	−0.893491	−	0.449081i	$-0.851752\pi$
$13$	−302.000	−1.78698	−0.893491	+	0.449081i	$0.851752\pi$
$14$	57.0000i	0.290816i
$15$	0	0
$16$	−95.0000	−0.371094
$17$	− 414.000i	− 1.43253i	−0.697830	−	0.716263i	$-0.745851\pi$
$17$	− 414.000i	− 1.43253i	0.697830	−	0.716263i	$-0.254149\pi$
$18$	0	0
$19$	−304.000	−0.842105	−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−304.000	−0.842105	−0.421053	+	0.907036i	$0.638339\pi$
$20$	0	0
$21$	0	0
$22$	−369.000	−0.762397
$23$	− 300.000i	− 0.567108i	−0.958956	−	0.283554i	$-0.908487\pi$
$23$	− 300.000i	− 0.567108i	0.958956	−	0.283554i	$-0.0915135\pi$
$24$	0	0
$25$	0	0
$26$	− 906.000i	− 1.34024i
$27$	0	0
$28$	133.000	0.169643
$29$	− 678.000i	− 0.806183i	−0.915160	−	0.403092i	$-0.867936\pi$
$29$	− 678.000i	− 0.806183i	0.915160	−	0.403092i	$-0.132064\pi$
$30$	0	0
$31$	239.000	0.248699	0.124350	−	0.992238i	$-0.460316\pi$
$31$	239.000	0.248699	0.124350	+	0.992238i	$0.460316\pi$
$32$	819.000i	0.799805i
$33$	0	0
$34$	1242.00	1.07439
$35$	0	0
$36$	0	0
$37$	−740.000	−0.540541	−0.270270	−	0.962784i	$-0.587113\pi$
$37$	−740.000	−0.540541	−0.270270	+	0.962784i	$0.587113\pi$
$38$	− 912.000i	− 0.631579i
$39$	0	0
$40$	0	0
$41$	228.000i	0.135634i	0.997698	+	0.0678168i	$0.0216033\pi$
$41$	228.000i	0.135634i	−0.997698	+	0.0678168i	$0.978397\pi$
$42$	0	0
$43$	982.000	0.531098	0.265549	−	0.964097i	$-0.414447\pi$
$43$	982.000	0.531098	0.265549	+	0.964097i	$0.414447\pi$
$44$	861.000i	0.444731i
$45$	0	0
$46$	900.000	0.425331
$47$	− 2166.00i	− 0.980534i	−0.871572	−	0.490267i	$-0.836899\pi$
$47$	− 2166.00i	− 0.980534i	0.871572	−	0.490267i	$-0.163101\pi$
$48$	0	0
$49$	−2040.00	−0.849646
$50$	0	0
$51$	0	0
$52$	−2114.00	−0.781805
$53$	1593.00i	0.567106i	0.958957	+	0.283553i	$0.0915131\pi$
$53$	1593.00i	0.567106i	−0.958957	+	0.283553i	$0.908487\pi$
$54$	0	0
$55$	0	0
$56$	1311.00i	0.418048i
$57$	0	0
$58$	2034.00	0.604637
$59$	− 2922.00i	− 0.839414i	−0.907660	−	0.419707i	$-0.862133\pi$
$59$	− 2922.00i	− 0.839414i	0.907660	−	0.419707i	$-0.137867\pi$
$60$	0	0
$61$	−316.000	−0.0849234	−0.0424617	−	0.999098i	$-0.513520\pi$
$61$	−316.000	−0.0849234	−0.0424617	+	0.999098i	$0.513520\pi$
$62$	717.000i	0.186524i
$63$	0	0
$64$	−3977.00	−0.970947
$65$	0	0
$66$	0	0
$67$	−4622.00	−1.02963	−0.514814	−	0.857302i	$-0.672139\pi$
$67$	−4622.00	−1.02963	−0.514814	+	0.857302i	$0.672139\pi$
$68$	− 2898.00i	− 0.626730i
$69$	0	0
$70$	0	0
$71$	1818.00i	0.360643i	0.983608	+	0.180321i	$0.0577138\pi$
$71$	1818.00i	0.360643i	−0.983608	+	0.180321i	$0.942286\pi$
$72$	0	0
$73$	3031.00	0.568775	0.284387	−	0.958709i	$-0.408210\pi$
$73$	3031.00	0.568775	0.284387	+	0.958709i	$0.408210\pi$
$74$	− 2220.00i	− 0.405405i
$75$	0	0
$76$	−2128.00	−0.368421
$77$	2337.00i	0.394164i
$78$	0	0
$79$	−10450.0	−1.67441	−0.837206	−	0.546888i	$-0.815812\pi$
$79$	−10450.0	−1.67441	−0.837206	+	0.546888i	$0.815812\pi$
$80$	0	0
$81$	0	0
$82$	−684.000	−0.101725
$83$	− 12633.0i	− 1.83379i	−0.399125	−	0.916897i	$-0.630686\pi$
$83$	− 12633.0i	− 1.83379i	0.399125	−	0.916897i	$-0.369314\pi$
$84$	0	0
$85$	0	0
$86$	2946.00i	0.398323i
$87$	0	0
$88$	−8487.00	−1.09595
$89$	− 7002.00i	− 0.883979i	−0.897020	−	0.441990i	$-0.854273\pi$
$89$	− 7002.00i	− 0.883979i	0.897020	−	0.441990i	$-0.145727\pi$
$90$	0	0
$91$	−5738.00	−0.692911
$92$	− 2100.00i	− 0.248110i
$93$	0	0
$94$	6498.00	0.735401
$95$	0	0
$96$	0	0
$97$	6517.00	0.692635	0.346317	−	0.938117i	$-0.387432\pi$
$97$	6517.00	0.692635	0.346317	+	0.938117i	$0.387432\pi$
$98$	− 6120.00i	− 0.637234i
$99$	0	0
$100$	0	0
$101$	5919.00i	0.580237i	0.956991	+	0.290119i	$0.0936948\pi$
$101$	5919.00i	0.580237i	−0.956991	+	0.290119i	$0.906305\pi$
$102$	0	0
$103$	7654.00	0.721463	0.360731	−	0.932670i	$-0.382527\pi$
$103$	7654.00	0.721463	0.360731	+	0.932670i	$0.382527\pi$
$104$	− 20838.0i	− 1.92659i
$105$	0	0
$106$	−4779.00	−0.425329
$107$	513.000i	0.0448074i	0.999749	+	0.0224037i	$0.00713192\pi$
$107$	513.000i	0.0448074i	−0.999749	+	0.0224037i	$0.992868\pi$
$108$	0	0
$109$	2324.00	0.195606	0.0978032	−	0.995206i	$-0.468818\pi$
$109$	2324.00	0.195606	0.0978032	+	0.995206i	$0.468818\pi$
$110$	0	0
$111$	0	0
$112$	−1805.00	−0.143893
$113$	4920.00i	0.385308i	0.981267	+	0.192654i	$0.0617095\pi$
$113$	4920.00i	0.385308i	−0.981267	+	0.192654i	$0.938290\pi$
$114$	0	0
$115$	0	0
$116$	− 4746.00i	− 0.352705i
$117$	0	0
$118$	8766.00	0.629560
$119$	− 7866.00i	− 0.555469i
$120$	0	0
$121$	−488.000	−0.0333311
$122$	− 948.000i	− 0.0636926i
$123$	0	0
$124$	1673.00	0.108806
$125$	0	0
$126$	0	0
$127$	−24995.0	−1.54969	−0.774847	−	0.632149i	$-0.782173\pi$
$127$	−24995.0	−1.54969	−0.774847	+	0.632149i	$0.782173\pi$
$128$	1173.00i	0.0715942i
$129$	0	0
$130$	0	0
$131$	28461.0i	1.65847i	0.558900	+	0.829235i	$0.311223\pi$
$131$	28461.0i	1.65847i	−0.558900	+	0.829235i	$0.688777\pi$
$132$	0	0
$133$	−5776.00	−0.326531
$134$	− 13866.0i	− 0.772221i
$135$	0	0
$136$	28566.0	1.54444
$137$	− 2454.00i	− 0.130748i	−0.997861	−	0.0653738i	$-0.979176\pi$
$137$	− 2454.00i	− 0.130748i	0.997861	−	0.0653738i	$-0.0208240\pi$
$138$	0	0
$139$	−11884.0	−0.615082	−0.307541	−	0.951535i	$-0.599506\pi$
$139$	−11884.0	−0.615082	−0.307541	+	0.951535i	$0.599506\pi$
$140$	0	0
$141$	0	0
$142$	−5454.00	−0.270482
$143$	− 37146.0i	− 1.81652i
$144$	0	0
$145$	0	0
$146$	9093.00i	0.426581i
$147$	0	0
$148$	−5180.00	−0.236486
$149$	− 21993.0i	− 0.990631i	−0.868713	−	0.495316i	$-0.835052\pi$
$149$	− 21993.0i	− 0.990631i	0.868713	−	0.495316i	$-0.164948\pi$
$150$	0	0
$151$	−2683.00	−0.117670	−0.0588351	−	0.998268i	$-0.518739\pi$
$151$	−2683.00	−0.117670	−0.0588351	+	0.998268i	$0.518739\pi$
$152$	− 20976.0i	− 0.907895i
$153$	0	0
$154$	−7011.00	−0.295623
$155$	0	0
$156$	0	0
$157$	32116.0	1.30293	0.651467	−	0.758677i	$-0.274154\pi$
$157$	32116.0	1.30293	0.651467	+	0.758677i	$0.274154\pi$
$158$	− 31350.0i	− 1.25581i
$159$	0	0
$160$	0	0
$161$	− 5700.00i	− 0.219899i
$162$	0	0
$163$	−22790.0	−0.857767	−0.428883	−	0.903360i	$-0.641093\pi$
$163$	−22790.0	−0.857767	−0.428883	+	0.903360i	$0.641093\pi$
$164$	1596.00i	0.0593397i
$165$	0	0
$166$	37899.0	1.37534
$167$	36078.0i	1.29363i	0.762648	+	0.646814i	$0.223899\pi$
$167$	36078.0i	1.29363i	−0.762648	+	0.646814i	$0.776101\pi$
$168$	0	0
$169$	62643.0	2.19331
$170$	0	0
$171$	0	0
$172$	6874.00	0.232355
$173$	− 19725.0i	− 0.659060i	−0.944145	−	0.329530i	$-0.893110\pi$
$173$	− 19725.0i	− 0.659060i	0.944145	−	0.329530i	$-0.106890\pi$
$174$	0	0
$175$	0	0
$176$	− 11685.0i	− 0.377228i
$177$	0	0
$178$	21006.0	0.662984
$179$	− 48915.0i	− 1.52664i	−0.646022	−	0.763319i	$-0.723569\pi$
$179$	− 48915.0i	− 1.52664i	0.646022	−	0.763319i	$-0.276431\pi$
$180$	0	0
$181$	−49552.0	−1.51253	−0.756265	−	0.654265i	$-0.772978\pi$
$181$	−49552.0	−1.51253	−0.756265	+	0.654265i	$0.772978\pi$
$182$	− 17214.0i	− 0.519684i
$183$	0	0
$184$	20700.0	0.611413
$185$	0	0
$186$	0	0
$187$	50922.0	1.45620
$188$	− 15162.0i	− 0.428984i
$189$	0	0
$190$	0	0
$191$	− 45390.0i	− 1.24421i	−0.782934	−	0.622105i	$-0.786278\pi$
$191$	− 45390.0i	− 1.24421i	0.782934	−	0.622105i	$-0.213722\pi$
$192$	0	0
$193$	−35447.0	−0.951623	−0.475811	−	0.879547i	$-0.657846\pi$
$193$	−35447.0	−0.951623	−0.475811	+	0.879547i	$0.657846\pi$
$194$	19551.0i	0.519476i
$195$	0	0
$196$	−14280.0	−0.371720
$197$	− 35739.0i	− 0.920895i	−0.887687	−	0.460447i	$-0.847689\pi$
$197$	− 35739.0i	− 0.920895i	0.887687	−	0.460447i	$-0.152311\pi$
$198$	0	0
$199$	−31255.0	−0.789248	−0.394624	−	0.918843i	$-0.629125\pi$
$199$	−31255.0	−0.789248	−0.394624	+	0.918843i	$0.629125\pi$
$200$	0	0
$201$	0	0
$202$	−17757.0	−0.435178
$203$	− 12882.0i	− 0.312602i
$204$	0	0
$205$	0	0
$206$	22962.0i	0.541097i
$207$	0	0
$208$	28690.0	0.663138
$209$	− 37392.0i	− 0.856024i
$210$	0	0
$211$	−15052.0	−0.338088	−0.169044	−	0.985609i	$-0.554068\pi$
$211$	−15052.0	−0.338088	−0.169044	+	0.985609i	$0.554068\pi$
$212$	11151.0i	0.248109i
$213$	0	0
$214$	−1539.00	−0.0336056
$215$	0	0
$216$	0	0
$217$	4541.00	0.0964344
$218$	6972.00i	0.146705i
$219$	0	0
$220$	0	0
$221$	125028.i	2.55990i
$222$	0	0
$223$	−50174.0	−1.00895	−0.504474	−	0.863427i	$-0.668314\pi$
$223$	−50174.0	−1.00895	−0.504474	+	0.863427i	$0.668314\pi$
$224$	15561.0i	0.310128i
$225$	0	0
$226$	−14760.0	−0.288981
$227$	− 19266.0i	− 0.373887i	−0.982371	−	0.186943i	$-0.940142\pi$
$227$	− 19266.0i	− 0.373887i	0.982371	−	0.186943i	$-0.0598581\pi$
$228$	0	0
$229$	34214.0	0.652428	0.326214	−	0.945296i	$-0.394227\pi$
$229$	34214.0	0.652428	0.326214	+	0.945296i	$0.394227\pi$
$230$	0	0
$231$	0	0
$232$	46782.0	0.869166
$233$	37386.0i	0.688648i	0.938851	+	0.344324i	$0.111892\pi$
$233$	37386.0i	0.688648i	−0.938851	+	0.344324i	$0.888108\pi$
$234$	0	0
$235$	0	0
$236$	− 20454.0i	− 0.367244i
$237$	0	0
$238$	23598.0	0.416602
$239$	− 61800.0i	− 1.08191i	−0.841050	−	0.540957i	$-0.818062\pi$
$239$	− 61800.0i	− 1.08191i	0.841050	−	0.540957i	$-0.181938\pi$
$240$	0	0
$241$	41390.0	0.712625	0.356313	−	0.934367i	$-0.384034\pi$
$241$	41390.0	0.712625	0.356313	+	0.934367i	$0.384034\pi$
$242$	− 1464.00i	− 0.0249983i
$243$	0	0
$244$	−2212.00	−0.0371540
$245$	0	0
$246$	0	0
$247$	91808.0	1.50483
$248$	16491.0i	0.268129i
$249$	0	0
$250$	0	0
$251$	− 82818.0i	− 1.31455i	−0.753651	−	0.657275i	$-0.771709\pi$
$251$	− 82818.0i	− 1.31455i	0.753651	−	0.657275i	$-0.228291\pi$
$252$	0	0
$253$	36900.0	0.576481
$254$	− 74985.0i	− 1.16227i
$255$	0	0
$256$	−67151.0	−1.02464
$257$	19590.0i	0.296598i	0.988943	+	0.148299i	$0.0473798\pi$
$257$	19590.0i	0.296598i	−0.988943	+	0.148299i	$0.952620\pi$
$258$	0	0
$259$	−14060.0	−0.209597
$260$	0	0
$261$	0	0
$262$	−85383.0	−1.24385
$263$	− 16692.0i	− 0.241322i	−0.992694	−	0.120661i	$-0.961499\pi$
$263$	− 16692.0i	− 0.241322i	0.992694	−	0.120661i	$-0.0385014\pi$
$264$	0	0
$265$	0	0
$266$	− 17328.0i	− 0.244898i
$267$	0	0
$268$	−32354.0	−0.450462
$269$	120906.i	1.67087i	0.549587	+	0.835436i	$0.314785\pi$
$269$	120906.i	1.67087i	−0.549587	+	0.835436i	$0.685215\pi$
$270$	0	0
$271$	73739.0	1.00406	0.502029	−	0.864851i	$-0.332587\pi$
$271$	73739.0	1.00406	0.502029	+	0.864851i	$0.332587\pi$
$272$	39330.0i	0.531601i
$273$	0	0
$274$	7362.00	0.0980606
$275$	0	0
$276$	0	0
$277$	−11996.0	−0.156342	−0.0781712	−	0.996940i	$-0.524908\pi$
$277$	−11996.0	−0.156342	−0.0781712	+	0.996940i	$0.524908\pi$
$278$	− 35652.0i	− 0.461312i
$279$	0	0
$280$	0	0
$281$	51126.0i	0.647484i	0.946145	+	0.323742i	$0.104941\pi$
$281$	51126.0i	0.647484i	−0.946145	+	0.323742i	$0.895059\pi$
$282$	0	0
$283$	1048.00	0.0130854	0.00654272	−	0.999979i	$-0.497917\pi$
$283$	1048.00	0.0130854	0.00654272	+	0.999979i	$0.497917\pi$
$284$	12726.0i	0.157781i
$285$	0	0
$286$	111438.	1.36239
$287$	4332.00i	0.0525926i
$288$	0	0
$289$	−87875.0	−1.05213
$290$	0	0
$291$	0	0
$292$	21217.0	0.248839
$293$	− 64182.0i	− 0.747615i	−0.927506	−	0.373807i	$-0.878052\pi$
$293$	− 64182.0i	− 0.747615i	0.927506	−	0.373807i	$-0.121948\pi$
$294$	0	0
$295$	0	0
$296$	− 51060.0i	− 0.582770i
$297$	0	0
$298$	65979.0	0.742973
$299$	90600.0i	1.01341i
$300$	0	0
$301$	18658.0	0.205936
$302$	− 8049.00i	− 0.0882527i
$303$	0	0
$304$	28880.0	0.312500
$305$	0	0
$306$	0	0
$307$	−154154.	−1.63560	−0.817802	−	0.575500i	$-0.804807\pi$
$307$	−154154.	−1.63560	−0.817802	+	0.575500i	$0.804807\pi$
$308$	16359.0i	0.172447i
$309$	0	0
$310$	0	0
$311$	94080.0i	0.972695i	0.873766	+	0.486347i	$0.161671\pi$
$311$	94080.0i	0.972695i	−0.873766	+	0.486347i	$0.838329\pi$
$312$	0	0
$313$	25903.0	0.264400	0.132200	−	0.991223i	$-0.457796\pi$
$313$	25903.0	0.264400	0.132200	+	0.991223i	$0.457796\pi$
$314$	96348.0i	0.977200i
$315$	0	0
$316$	−73150.0	−0.732555
$317$	96843.0i	0.963717i	0.876249	+	0.481859i	$0.160038\pi$
$317$	96843.0i	0.963717i	−0.876249	+	0.481859i	$0.839962\pi$
$318$	0	0
$319$	83394.0	0.819508
$320$	0	0
$321$	0	0
$322$	17100.0	0.164924
$323$	125856.i	1.20634i
$324$	0	0
$325$	0	0
$326$	− 68370.0i	− 0.643325i
$327$	0	0
$328$	−15732.0	−0.146230
$329$	− 41154.0i	− 0.380207i
$330$	0	0
$331$	−164854.	−1.50468	−0.752339	−	0.658776i	$-0.771074\pi$
$331$	−164854.	−1.50468	−0.752339	+	0.658776i	$0.771074\pi$
$332$	− 88431.0i	− 0.802284i
$333$	0	0
$334$	−108234.	−0.970221
$335$	0	0
$336$	0	0
$337$	−148694.	−1.30928	−0.654642	−	0.755939i	$-0.727180\pi$
$337$	−148694.	−1.30928	−0.654642	+	0.755939i	$0.727180\pi$
$338$	187929.i	1.64498i
$339$	0	0
$340$	0	0
$341$	29397.0i	0.252810i
$342$	0	0
$343$	−84379.0	−0.717210
$344$	67758.0i	0.572590i
$345$	0	0
$346$	59175.0	0.494295
$347$	107673.i	0.894227i	0.894477	+	0.447114i	$0.147548\pi$
$347$	107673.i	0.894227i	−0.894477	+	0.447114i	$0.852452\pi$
$348$	0	0
$349$	127520.	1.04695	0.523477	−	0.852040i	$-0.324635\pi$
$349$	127520.	1.04695	0.523477	+	0.852040i	$0.324635\pi$
$350$	0	0
$351$	0	0
$352$	−100737.	−0.813025
$353$	142104.i	1.14040i	0.821506	+	0.570200i	$0.193134\pi$
$353$	142104.i	1.14040i	−0.821506	+	0.570200i	$0.806866\pi$
$354$	0	0
$355$	0	0
$356$	− 49014.0i	− 0.386741i
$357$	0	0
$358$	146745.	1.14498
$359$	− 19422.0i	− 0.150697i	−0.997157	−	0.0753486i	$-0.975993\pi$
$359$	− 19422.0i	− 0.150697i	0.997157	−	0.0753486i	$-0.0240069\pi$
$360$	0	0
$361$	−37905.0	−0.290859
$362$	− 148656.i	− 1.13440i
$363$	0	0
$364$	−40166.0	−0.303149
$365$	0	0
$366$	0	0
$367$	151345.	1.12366	0.561831	−	0.827252i	$-0.310097\pi$
$367$	151345.	1.12366	0.561831	+	0.827252i	$0.310097\pi$
$368$	28500.0i	0.210450i
$369$	0	0
$370$	0	0
$371$	30267.0i	0.219898i
$372$	0	0
$373$	−237506.	−1.70709	−0.853546	−	0.521018i	$-0.825553\pi$
$373$	−237506.	−1.70709	−0.853546	+	0.521018i	$0.825553\pi$
$374$	152766.i	1.09215i
$375$	0	0
$376$	149454.	1.05714
$377$	204756.i	1.44063i
$378$	0	0
$379$	−261952.	−1.82366	−0.911829	−	0.410571i	$-0.865330\pi$
$379$	−261952.	−1.82366	−0.911829	+	0.410571i	$0.865330\pi$
$380$	0	0
$381$	0	0
$382$	136170.	0.933157
$383$	87162.0i	0.594196i	0.954847	+	0.297098i	$0.0960188\pi$
$383$	87162.0i	0.594196i	−0.954847	+	0.297098i	$0.903981\pi$
$384$	0	0
$385$	0	0
$386$	− 106341.i	− 0.713717i
$387$	0	0
$388$	45619.0	0.303028
$389$	239343.i	1.58169i	0.612016	+	0.790845i	$0.290359\pi$
$389$	239343.i	1.58169i	−0.612016	+	0.790845i	$0.709641\pi$
$390$	0	0
$391$	−124200.	−0.812397
$392$	− 140760.i	− 0.916025i
$393$	0	0
$394$	107217.	0.690671
$395$	0	0
$396$	0	0
$397$	−217154.	−1.37780	−0.688901	−	0.724855i	$-0.741907\pi$
$397$	−217154.	−1.37780	−0.688901	+	0.724855i	$0.741907\pi$
$398$	− 93765.0i	− 0.591936i
$399$	0	0
$400$	0	0
$401$	− 256200.i	− 1.59327i	−0.604458	−	0.796637i	$-0.706610\pi$
$401$	− 256200.i	− 1.59327i	0.604458	−	0.796637i	$-0.293390\pi$
$402$	0	0
$403$	−72178.0	−0.444421
$404$	41433.0i	0.253854i
$405$	0	0
$406$	38646.0	0.234451
$407$	− 91020.0i	− 0.549475i
$408$	0	0
$409$	−199291.	−1.19135	−0.595677	−	0.803224i	$-0.703116\pi$
$409$	−199291.	−1.19135	−0.595677	+	0.803224i	$0.703116\pi$
$410$	0	0
$411$	0	0
$412$	53578.0	0.315640
$413$	− 55518.0i	− 0.325487i
$414$	0	0
$415$	0	0
$416$	− 247338.i	− 1.42924i
$417$	0	0
$418$	112176.	0.642018
$419$	251274.i	1.43126i	0.698478	+	0.715632i	$0.253861\pi$
$419$	251274.i	1.43126i	−0.698478	+	0.715632i	$0.746139\pi$
$420$	0	0
$421$	−30412.0	−0.171586	−0.0857928	−	0.996313i	$-0.527342\pi$
$421$	−30412.0	−0.171586	−0.0857928	+	0.996313i	$0.527342\pi$
$422$	− 45156.0i	− 0.253566i
$423$	0	0
$424$	−109917.	−0.611411
$425$	0	0
$426$	0	0
$427$	−6004.00	−0.0329295
$428$	3591.00i	0.0196032i
$429$	0	0
$430$	0	0
$431$	161730.i	0.870635i	0.900277	+	0.435317i	$0.143364\pi$
$431$	161730.i	0.870635i	−0.900277	+	0.435317i	$0.856636\pi$
$432$	0	0
$433$	213541.	1.13895	0.569476	−	0.822008i	$-0.307146\pi$
$433$	213541.	1.13895	0.569476	+	0.822008i	$0.307146\pi$
$434$	13623.0i	0.0723258i
$435$	0	0
$436$	16268.0	0.0855778
$437$	91200.0i	0.477564i
$438$	0	0
$439$	66725.0	0.346226	0.173113	−	0.984902i	$-0.444617\pi$
$439$	66725.0	0.346226	0.173113	+	0.984902i	$0.444617\pi$
$440$	0	0
$441$	0	0
$442$	−375084.	−1.91992
$443$	274170.i	1.39705i	0.715585	+	0.698526i	$0.246160\pi$
$443$	274170.i	1.39705i	−0.715585	+	0.698526i	$0.753840\pi$
$444$	0	0
$445$	0	0
$446$	− 150522.i	− 0.756711i
$447$	0	0
$448$	−75563.0	−0.376490
$449$	− 233784.i	− 1.15964i	−0.814746	−	0.579819i	$-0.803123\pi$
$449$	− 233784.i	− 1.15964i	0.814746	−	0.579819i	$-0.196877\pi$
$450$	0	0
$451$	−28044.0	−0.137875
$452$	34440.0i	0.168572i
$453$	0	0
$454$	57798.0	0.280415
$455$	0	0
$456$	0	0
$457$	90667.0	0.434127	0.217064	−	0.976157i	$-0.430352\pi$
$457$	90667.0	0.434127	0.217064	+	0.976157i	$0.430352\pi$
$458$	102642.i	0.489321i
$459$	0	0
$460$	0	0
$461$	201957.i	0.950292i	0.879907	+	0.475146i	$0.157605\pi$
$461$	201957.i	0.950292i	−0.879907	+	0.475146i	$0.842395\pi$
$462$	0	0
$463$	323977.	1.51131	0.755653	−	0.654973i	$-0.227320\pi$
$463$	323977.	1.51131	0.755653	+	0.654973i	$0.227320\pi$
$464$	64410.0i	0.299170i
$465$	0	0
$466$	−112158.	−0.516486
$467$	− 76941.0i	− 0.352796i	−0.984319	−	0.176398i	$-0.943555\pi$
$467$	− 76941.0i	− 0.352796i	0.984319	−	0.176398i	$-0.0564446\pi$
$468$	0	0
$469$	−87818.0	−0.399244
$470$	0	0
$471$	0	0
$472$	201618.	0.904993
$473$	120786.i	0.539876i
$474$	0	0
$475$	0	0
$476$	− 55062.0i	− 0.243018i
$477$	0	0
$478$	185400.	0.811435
$479$	193218.i	0.842125i	0.907032	+	0.421062i	$0.138343\pi$
$479$	193218.i	0.842125i	−0.907032	+	0.421062i	$0.861657\pi$
$480$	0	0
$481$	223480.	0.965936
$482$	124170.i	0.534469i
$483$	0	0
$484$	−3416.00	−0.0145823
$485$	0	0
$486$	0	0
$487$	34882.0	0.147077	0.0735383	−	0.997292i	$-0.476571\pi$
$487$	34882.0	0.147077	0.0735383	+	0.997292i	$0.476571\pi$
$488$	− 21804.0i	− 0.0915580i
$489$	0	0
$490$	0	0
$491$	217047.i	0.900307i	0.892951	+	0.450154i	$0.148631\pi$
$491$	217047.i	0.900307i	−0.892951	+	0.450154i	$0.851369\pi$
$492$	0	0
$493$	−280692.	−1.15488
$494$	275424.i	1.12862i
$495$	0	0
$496$	−22705.0	−0.0922907
$497$	34542.0i	0.139841i
$498$	0	0
$499$	464810.	1.86670	0.933350	−	0.358969i	$-0.116871\pi$
$499$	464810.	1.86670	0.933350	+	0.358969i	$0.116871\pi$
$500$	0	0
$501$	0	0
$502$	248454.	0.985913
$503$	167580.i	0.662348i	0.943570	+	0.331174i	$0.107445\pi$
$503$	167580.i	0.662348i	−0.943570	+	0.331174i	$0.892555\pi$
$504$	0	0
$505$	0	0
$506$	110700.i	0.432361i
$507$	0	0
$508$	−174965.	−0.677991
$509$	35697.0i	0.137783i	0.997624	+	0.0688916i	$0.0219463\pi$
$509$	35697.0i	0.137783i	−0.997624	+	0.0688916i	$0.978054\pi$
$510$	0	0
$511$	57589.0	0.220545
$512$	− 182685.i	− 0.696888i
$513$	0	0
$514$	−58770.0	−0.222448
$515$	0	0
$516$	0	0
$517$	266418.	0.996741
$518$	− 42180.0i	− 0.157198i
$519$	0	0
$520$	0	0
$521$	− 42750.0i	− 0.157493i	−0.996895	−	0.0787464i	$-0.974908\pi$
$521$	− 42750.0i	− 0.157493i	0.996895	−	0.0787464i	$-0.0250917\pi$
$522$	0	0
$523$	176434.	0.645028	0.322514	−	0.946565i	$-0.395472\pi$
$523$	176434.	0.645028	0.322514	+	0.946565i	$0.395472\pi$
$524$	199227.i	0.725581i
$525$	0	0
$526$	50076.0	0.180991
$527$	− 98946.0i	− 0.356268i
$528$	0	0
$529$	189841.	0.678389
$530$	0	0
$531$	0	0
$532$	−40432.0	−0.142857
$533$	− 68856.0i	− 0.242375i
$534$	0	0
$535$	0	0
$536$	− 318918.i	− 1.11007i
$537$	0	0
$538$	−362718.	−1.25315
$539$	− 250920.i	− 0.863690i
$540$	0	0
$541$	−323836.	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$541$	−323836.	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$542$	221217.i	0.753043i
$543$	0	0
$544$	339066.	1.14574
$545$	0	0
$546$	0	0
$547$	223390.	0.746602	0.373301	−	0.927710i	$-0.378226\pi$
$547$	223390.	0.746602	0.373301	+	0.927710i	$0.378226\pi$
$548$	− 17178.0i	− 0.0572020i
$549$	0	0
$550$	0	0
$551$	206112.i	0.678891i
$552$	0	0
$553$	−198550.	−0.649261
$554$	− 35988.0i	− 0.117257i
$555$	0	0
$556$	−83188.0	−0.269098
$557$	585027.i	1.88567i	0.333261	+	0.942835i	$0.391851\pi$
$557$	585027.i	1.88567i	−0.333261	+	0.942835i	$0.608149\pi$
$558$	0	0
$559$	−296564.	−0.949063
$560$	0	0
$561$	0	0
$562$	−153378.	−0.485613
$563$	84075.0i	0.265247i	0.991167	+	0.132623i	$0.0423401\pi$
$563$	84075.0i	0.265247i	−0.991167	+	0.132623i	$0.957660\pi$
$564$	0	0
$565$	0	0
$566$	3144.00i	0.00981408i
$567$	0	0
$568$	−125442.	−0.388818
$569$	− 637392.i	− 1.96871i	−0.176192	−	0.984356i	$-0.556378\pi$
$569$	− 637392.i	− 1.96871i	0.176192	−	0.984356i	$-0.443622\pi$
$570$	0	0
$571$	80726.0	0.247595	0.123797	−	0.992308i	$-0.460493\pi$
$571$	80726.0	0.247595	0.123797	+	0.992308i	$0.460493\pi$
$572$	− 260022.i	− 0.794727i
$573$	0	0
$574$	−12996.0	−0.0394445
$575$	0	0
$576$	0	0
$577$	−261182.	−0.784498	−0.392249	−	0.919859i	$-0.628303\pi$
$577$	−261182.	−0.784498	−0.392249	+	0.919859i	$0.628303\pi$
$578$	− 263625.i	− 0.789098i
$579$	0	0
$580$	0	0
$581$	− 240027.i	− 0.711063i
$582$	0	0
$583$	−195939.	−0.576479
$584$	209139.i	0.613210i
$585$	0	0
$586$	192546.	0.560711
$587$	391305.i	1.13564i	0.823154	+	0.567818i	$0.192212\pi$
$587$	391305.i	1.13564i	−0.823154	+	0.567818i	$0.807788\pi$
$588$	0	0
$589$	−72656.0	−0.209431
$590$	0	0
$591$	0	0
$592$	70300.0	0.200591
$593$	− 302670.i	− 0.860716i	−0.902658	−	0.430358i	$-0.858387\pi$
$593$	− 302670.i	− 0.860716i	0.902658	−	0.430358i	$-0.141613\pi$
$594$	0	0
$595$	0	0
$596$	− 153951.i	− 0.433401i
$597$	0	0
$598$	−271800.	−0.760059
$599$	− 291498.i	− 0.812422i	−0.913779	−	0.406211i	$-0.866850\pi$
$599$	− 291498.i	− 0.812422i	0.913779	−	0.406211i	$-0.133150\pi$
$600$	0	0
$601$	402173.	1.11343	0.556716	−	0.830703i	$-0.312061\pi$
$601$	402173.	1.11343	0.556716	+	0.830703i	$0.312061\pi$
$602$	55974.0i	0.154452i
$603$	0	0
$604$	−18781.0	−0.0514807
$605$	0	0
$606$	0	0
$607$	378670.	1.02774	0.513870	−	0.857868i	$-0.328211\pi$
$607$	378670.	1.02774	0.513870	+	0.857868i	$0.328211\pi$
$608$	− 248976.i	− 0.673520i
$609$	0	0
$610$	0	0
$611$	654132.i	1.75220i
$612$	0	0
$613$	−287570.	−0.765284	−0.382642	−	0.923897i	$-0.624986\pi$
$613$	−287570.	−0.765284	−0.382642	+	0.923897i	$0.624986\pi$
$614$	− 462462.i	− 1.22670i
$615$	0	0
$616$	−161253.	−0.424958
$617$	− 576264.i	− 1.51374i	−0.653566	−	0.756870i	$-0.726728\pi$
$617$	− 576264.i	− 1.51374i	0.653566	−	0.756870i	$-0.273272\pi$
$618$	0	0
$619$	223262.	0.582685	0.291342	−	0.956619i	$-0.405898\pi$
$619$	223262.	0.582685	0.291342	+	0.956619i	$0.405898\pi$
$620$	0	0
$621$	0	0
$622$	−282240.	−0.729521
$623$	− 133038.i	− 0.342767i
$624$	0	0
$625$	0	0
$626$	77709.0i	0.198300i
$627$	0	0
$628$	224812.	0.570033
$629$	306360.i	0.774338i
$630$	0	0
$631$	43373.0	0.108933	0.0544667	−	0.998516i	$-0.482654\pi$
$631$	43373.0	0.108933	0.0544667	+	0.998516i	$0.482654\pi$
$632$	− 721050.i	− 1.80522i
$633$	0	0
$634$	−290529.	−0.722788
$635$	0	0
$636$	0	0
$637$	616080.	1.51830
$638$	250182.i	0.614631i
$639$	0	0
$640$	0	0
$641$	423420.i	1.03052i	0.857035	+	0.515259i	$0.172304\pi$
$641$	423420.i	1.03052i	−0.857035	+	0.515259i	$0.827696\pi$
$642$	0	0
$643$	546088.	1.32081	0.660406	−	0.750909i	$-0.270384\pi$
$643$	546088.	1.32081	0.660406	+	0.750909i	$0.270384\pi$
$644$	− 39900.0i	− 0.0962058i
$645$	0	0
$646$	−377568.	−0.904753
$647$	418932.i	1.00077i	0.865803	+	0.500386i	$0.166808\pi$
$647$	418932.i	1.00077i	−0.865803	+	0.500386i	$0.833192\pi$
$648$	0	0
$649$	359406.	0.853289
$650$	0	0
$651$	0	0
$652$	−159530.	−0.375273
$653$	− 703209.i	− 1.64914i	−0.565758	−	0.824571i	$-0.691417\pi$
$653$	− 703209.i	− 1.64914i	0.565758	−	0.824571i	$-0.308583\pi$
$654$	0	0
$655$	0	0
$656$	− 21660.0i	− 0.0503328i
$657$	0	0
$658$	123462.	0.285155
$659$	102021.i	0.234919i	0.993078	+	0.117460i	$0.0374751\pi$
$659$	102021.i	0.234919i	−0.993078	+	0.117460i	$0.962525\pi$
$660$	0	0
$661$	230720.	0.528059	0.264029	−	0.964515i	$-0.414948\pi$
$661$	230720.	0.528059	0.264029	+	0.964515i	$0.414948\pi$
$662$	− 494562.i	− 1.12851i
$663$	0	0
$664$	871677.	1.97706
$665$	0	0
$666$	0	0
$667$	−203400.	−0.457193
$668$	252546.i	0.565962i
$669$	0	0
$670$	0	0
$671$	− 38868.0i	− 0.0863271i
$672$	0	0
$673$	469369.	1.03630	0.518149	−	0.855291i	$-0.326621\pi$
$673$	469369.	1.03630	0.518149	+	0.855291i	$0.326621\pi$
$674$	− 446082.i	− 0.981963i
$675$	0	0
$676$	438501.	0.959571
$677$	343146.i	0.748689i	0.927290	+	0.374345i	$0.122132\pi$
$677$	343146.i	0.748689i	−0.927290	+	0.374345i	$0.877868\pi$
$678$	0	0
$679$	123823.	0.268573
$680$	0	0
$681$	0	0
$682$	−88191.0	−0.189608
$683$	− 24642.0i	− 0.0528244i	−0.999651	−	0.0264122i	$-0.991592\pi$
$683$	− 24642.0i	− 0.0528244i	0.999651	−	0.0264122i	$-0.00840824\pi$
$684$	0	0
$685$	0	0
$686$	− 253137.i	− 0.537907i
$687$	0	0
$688$	−93290.0	−0.197087
$689$	− 481086.i	− 1.01341i
$690$	0	0
$691$	−266500.	−0.558137	−0.279069	−	0.960271i	$-0.590026\pi$
$691$	−266500.	−0.558137	−0.279069	+	0.960271i	$0.590026\pi$
$692$	− 138075.i	− 0.288339i
$693$	0	0
$694$	−323019.	−0.670670
$695$	0	0
$696$	0	0
$697$	94392.0	0.194299
$698$	382560.i	0.785215i
$699$	0	0
$700$	0	0
$701$	690309.i	1.40478i	0.711794	+	0.702389i	$0.247883\pi$
$701$	690309.i	1.40478i	−0.711794	+	0.702389i	$0.752117\pi$
$702$	0	0
$703$	224960.	0.455192
$704$	− 489171.i	− 0.986996i
$705$	0	0
$706$	−426312.	−0.855299
$707$	112461.i	0.224990i
$708$	0	0
$709$	−105184.	−0.209246	−0.104623	−	0.994512i	$-0.533364\pi$
$709$	−105184.	−0.209246	−0.104623	+	0.994512i	$0.533364\pi$
$710$	0	0
$711$	0	0
$712$	483138.	0.953040
$713$	− 71700.0i	− 0.141039i
$714$	0	0
$715$	0	0
$716$	− 342405.i	− 0.667904i
$717$	0	0
$718$	58266.0	0.113023
$719$	− 704988.i	− 1.36372i	−0.731485	−	0.681858i	$-0.761172\pi$
$719$	− 704988.i	− 1.36372i	0.731485	−	0.681858i	$-0.238828\pi$
$720$	0	0
$721$	145426.	0.279751
$722$	− 113715.i	− 0.218144i
$723$	0	0
$724$	−346864.	−0.661732
$725$	0	0
$726$	0	0
$727$	−126089.	−0.238566	−0.119283	−	0.992860i	$-0.538060\pi$
$727$	−126089.	−0.238566	−0.119283	+	0.992860i	$0.538060\pi$
$728$	− 395922.i	− 0.747045i
$729$	0	0
$730$	0	0
$731$	− 406548.i	− 0.760812i
$732$	0	0
$733$	−97736.0	−0.181906	−0.0909529	−	0.995855i	$-0.528991\pi$
$733$	−97736.0	−0.181906	−0.0909529	+	0.995855i	$0.528991\pi$
$734$	454035.i	0.842747i
$735$	0	0
$736$	245700.	0.453575
$737$	− 568506.i	− 1.04665i
$738$	0	0
$739$	−857158.	−1.56954	−0.784769	−	0.619788i	$-0.787219\pi$
$739$	−857158.	−1.56954	−0.784769	+	0.619788i	$0.787219\pi$
$740$	0	0
$741$	0	0
$742$	−90801.0	−0.164924
$743$	− 909966.i	− 1.64834i	−0.566340	−	0.824171i	$-0.691641\pi$
$743$	− 909966.i	− 1.64834i	0.566340	−	0.824171i	$-0.308359\pi$
$744$	0	0
$745$	0	0
$746$	− 712518.i	− 1.28032i
$747$	0	0
$748$	356454.	0.637089
$749$	9747.00i	0.0173743i
$750$	0	0
$751$	61223.0	0.108551	0.0542756	−	0.998526i	$-0.482715\pi$
$751$	61223.0	0.108551	0.0542756	+	0.998526i	$0.482715\pi$
$752$	205770.i	0.363870i
$753$	0	0
$754$	−614268.	−1.08048
$755$	0	0
$756$	0	0
$757$	−782570.	−1.36562	−0.682812	−	0.730594i	$-0.739243\pi$
$757$	−782570.	−1.36562	−0.682812	+	0.730594i	$0.739243\pi$
$758$	− 785856.i	− 1.36774i
$759$	0	0
$760$	0	0
$761$	701400.i	1.21115i	0.795790	+	0.605573i	$0.207056\pi$
$761$	701400.i	1.21115i	−0.795790	+	0.605573i	$0.792944\pi$
$762$	0	0
$763$	44156.0	0.0758474
$764$	− 317730.i	− 0.544342i
$765$	0	0
$766$	−261486.	−0.445647
$767$	882444.i	1.50002i
$768$	0	0
$769$	−85045.0	−0.143812	−0.0719062	−	0.997411i	$-0.522908\pi$
$769$	−85045.0	−0.143812	−0.0719062	+	0.997411i	$0.522908\pi$
$770$	0	0
$771$	0	0
$772$	−248129.	−0.416335
$773$	− 643122.i	− 1.07630i	−0.842848	−	0.538151i	$-0.819123\pi$
$773$	− 643122.i	− 1.07630i	0.842848	−	0.538151i	$-0.180877\pi$
$774$	0	0
$775$	0	0
$776$	449673.i	0.746747i
$777$	0	0
$778$	−718029.	−1.18627
$779$	− 69312.0i	− 0.114218i
$780$	0	0
$781$	−223614.	−0.366604
$782$	− 372600.i	− 0.609297i
$783$	0	0
$784$	193800.	0.315298
$785$	0	0
$786$	0	0
$787$	−1.06855e6	−1.72522	−0.862610	−	0.505869i	$-0.831172\pi$
$787$	−1.06855e6	−1.72522	−0.862610	+	0.505869i	$0.831172\pi$
$788$	− 250173.i	− 0.402891i
$789$	0	0
$790$	0	0
$791$	93480.0i	0.149405i
$792$	0	0
$793$	95432.0	0.151757
$794$	− 651462.i	− 1.03335i
$795$	0	0
$796$	−218785.	−0.345296
$797$	538935.i	0.848437i	0.905560	+	0.424219i	$0.139451\pi$
$797$	538935.i	0.848437i	−0.905560	+	0.424219i	$0.860549\pi$
$798$	0	0
$799$	−896724.	−1.40464
$800$	0	0
$801$	0	0
$802$	768600.	1.19496
$803$	372813.i	0.578176i
$804$	0	0
$805$	0	0
$806$	− 216534.i	− 0.333316i
$807$	0	0
$808$	−408411.	−0.625568
$809$	− 459594.i	− 0.702227i	−0.936333	−	0.351113i	$-0.885803\pi$
$809$	− 459594.i	− 0.702227i	0.936333	−	0.351113i	$-0.114197\pi$
$810$	0	0
$811$	−961360.	−1.46165	−0.730827	−	0.682563i	$-0.760865\pi$
$811$	−961360.	−1.46165	−0.730827	+	0.682563i	$0.760865\pi$
$812$	− 90174.0i	− 0.136763i
$813$	0	0
$814$	273060.	0.412106
$815$	0	0
$816$	0	0
$817$	−298528.	−0.447240
$818$	− 597873.i	− 0.893516i
$819$	0	0
$820$	0	0
$821$	105666.i	0.156765i	0.996923	+	0.0783825i	$0.0249755\pi$
$821$	105666.i	0.156765i	−0.996923	+	0.0783825i	$0.975024\pi$
$822$	0	0
$823$	493555.	0.728678	0.364339	−	0.931266i	$-0.381295\pi$
$823$	493555.	0.728678	0.364339	+	0.931266i	$0.381295\pi$
$824$	528126.i	0.777827i
$825$	0	0
$826$	166554.	0.244115
$827$	− 192870.i	− 0.282003i	−0.990009	−	0.141001i	$-0.954968\pi$
$827$	− 192870.i	− 0.282003i	0.990009	−	0.141001i	$-0.0450322\pi$
$828$	0	0
$829$	577226.	0.839918	0.419959	−	0.907543i	$-0.362044\pi$
$829$	577226.	0.839918	0.419959	+	0.907543i	$0.362044\pi$
$830$	0	0
$831$	0	0
$832$	1.20105e6	1.73507
$833$	844560.i	1.21714i
$834$	0	0
$835$	0	0
$836$	− 261744.i	− 0.374511i
$837$	0	0
$838$	−753822.	−1.07345
$839$	− 70986.0i	− 0.100844i	−0.998728	−	0.0504219i	$-0.983943\pi$
$839$	− 70986.0i	− 0.100844i	0.998728	−	0.0504219i	$-0.0160566\pi$
$840$	0	0
$841$	247597.	0.350069
$842$	− 91236.0i	− 0.128689i
$843$	0	0
$844$	−105364.	−0.147913
$845$	0	0
$846$	0	0
$847$	−9272.00	−0.0129243
$848$	− 151335.i	− 0.210449i
$849$	0	0
$850$	0	0
$851$	222000.i	0.306545i
$852$	0	0
$853$	−81206.0	−0.111607	−0.0558033	−	0.998442i	$-0.517772\pi$
$853$	−81206.0	−0.111607	−0.0558033	+	0.998442i	$0.517772\pi$
$854$	− 18012.0i	− 0.0246971i
$855$	0	0
$856$	−35397.0	−0.0483080
$857$	− 1.08044e6i	− 1.47109i	−0.677478	−	0.735543i	$-0.736927\pi$
$857$	− 1.08044e6i	− 1.47109i	0.677478	−	0.735543i	$-0.263073\pi$
$858$	0	0
$859$	503348.	0.682153	0.341077	−	0.940035i	$-0.389208\pi$
$859$	503348.	0.682153	0.341077	+	0.940035i	$0.389208\pi$
$860$	0	0
$861$	0	0
$862$	−485190.	−0.652976
$863$	548100.i	0.735933i	0.929839	+	0.367966i	$0.119946\pi$
$863$	548100.i	0.735933i	−0.929839	+	0.367966i	$0.880054\pi$
$864$	0	0
$865$	0	0
$866$	640623.i	0.854214i
$867$	0	0
$868$	31787.0	0.0421901
$869$	− 1.28535e6i	− 1.70209i
$870$	0	0
$871$	1.39584e6	1.83993
$872$	160356.i	0.210888i
$873$	0	0
$874$	−273600.	−0.358173
$875$	0	0
$876$	0	0
$877$	−700034.	−0.910165	−0.455082	−	0.890449i	$-0.650390\pi$
$877$	−700034.	−0.910165	−0.455082	+	0.890449i	$0.650390\pi$
$878$	200175.i	0.259669i
$879$	0	0
$880$	0	0
$881$	806634.i	1.03926i	0.854391	+	0.519631i	$0.173930\pi$
$881$	806634.i	1.03926i	−0.854391	+	0.519631i	$0.826070\pi$
$882$	0	0
$883$	−342704.	−0.439539	−0.219770	−	0.975552i	$-0.570531\pi$
$883$	−342704.	−0.439539	−0.219770	+	0.975552i	$0.570531\pi$
$884$	875196.i	1.11996i
$885$	0	0
$886$	−822510.	−1.04779
$887$	− 16122.0i	− 0.0204914i	−0.999948	−	0.0102457i	$-0.996739\pi$
$887$	− 16122.0i	− 0.0204914i	0.999948	−	0.0102457i	$-0.00326137\pi$
$888$	0	0
$889$	−474905.	−0.600901
$890$	0	0
$891$	0	0
$892$	−351218.	−0.441415
$893$	658464.i	0.825713i
$894$	0	0
$895$	0	0
$896$	22287.0i	0.0277610i
$897$	0	0
$898$	701352.	0.869728
$899$	− 162042.i	− 0.200497i
$900$	0	0
$901$	659502.	0.812394
$902$	− 84132.0i	− 0.103407i
$903$	0	0
$904$	−339480.	−0.415410
$905$	0	0
$906$	0	0
$907$	−1.56950e6	−1.90786	−0.953931	−	0.300028i	$-0.903004\pi$
$907$	−1.56950e6	−1.90786	−0.953931	+	0.300028i	$0.903004\pi$
$908$	− 134862.i	− 0.163575i
$909$	0	0
$910$	0	0
$911$	− 500898.i	− 0.603549i	−0.953379	−	0.301775i	$-0.902421\pi$
$911$	− 500898.i	− 0.603549i	0.953379	−	0.301775i	$-0.0975790\pi$
$912$	0	0
$913$	1.55386e6	1.86410
$914$	272001.i	0.325595i
$915$	0	0
$916$	239498.	0.285437
$917$	540759.i	0.643080i
$918$	0	0
$919$	1.03715e6	1.22804	0.614019	−	0.789291i	$-0.289552\pi$
$919$	1.03715e6	1.22804	0.614019	+	0.789291i	$0.289552\pi$
$920$	0	0
$921$	0	0
$922$	−605871.	−0.712719
$923$	− 549036.i	− 0.644462i
$924$	0	0
$925$	0	0
$926$	971931.i	1.13348i
$927$	0	0
$928$	555282.	0.644789
$929$	128076.i	0.148401i	0.997243	+	0.0742004i	$0.0236405\pi$
$929$	128076.i	0.148401i	−0.997243	+	0.0742004i	$0.976360\pi$
$930$	0	0
$931$	620160.	0.715491
$932$	261702.i	0.301283i
$933$	0	0
$934$	230823.	0.264597
$935$	0	0
$936$	0	0
$937$	879451.	1.00169	0.500844	−	0.865538i	$-0.333023\pi$
$937$	879451.	1.00169	0.500844	+	0.865538i	$0.333023\pi$
$938$	− 263454.i	− 0.299433i
$939$	0	0
$940$	0	0
$941$	718257.i	0.811149i	0.914062	+	0.405574i	$0.132929\pi$
$941$	718257.i	0.811149i	−0.914062	+	0.405574i	$0.867071\pi$
$942$	0	0
$943$	68400.0	0.0769188
$944$	277590.i	0.311501i
$945$	0	0
$946$	−362358.	−0.404907
$947$	− 73005.0i	− 0.0814053i	−0.999171	−	0.0407026i	$-0.987040\pi$
$947$	− 73005.0i	− 0.0814053i	0.999171	−	0.0407026i	$-0.0129596\pi$
$948$	0	0
$949$	−915362.	−1.01639
$950$	0	0
$951$	0	0
$952$	542754.	0.598865
$953$	− 309168.i	− 0.340415i	−0.985408	−	0.170208i	$-0.945556\pi$
$953$	− 309168.i	− 0.340415i	0.985408	−	0.170208i	$-0.0544438\pi$
$954$	0	0
$955$	0	0
$956$	− 432600.i	− 0.473337i
$957$	0	0
$958$	−579654.	−0.631594
$959$	− 46626.0i	− 0.0506980i
$960$	0	0
$961$	−866400.	−0.938149
$962$	670440.i	0.724452i
$963$	0	0
$964$	289730.	0.311774
$965$	0	0
$966$	0	0
$967$	366187.	0.391607	0.195803	−	0.980643i	$-0.437269\pi$
$967$	366187.	0.391607	0.195803	+	0.980643i	$0.437269\pi$
$968$	− 33672.0i	− 0.0359350i
$969$	0	0
$970$	0	0
$971$	− 1.43410e6i	− 1.52105i	−0.649311	−	0.760523i	$-0.724943\pi$
$971$	− 1.43410e6i	− 1.52105i	0.649311	−	0.760523i	$-0.275057\pi$
$972$	0	0
$973$	−225796.	−0.238501
$974$	104646.i	0.110307i
$975$	0	0
$976$	30020.0	0.0315145
$977$	311802.i	0.326655i	0.986572	+	0.163328i	$0.0522228\pi$
$977$	311802.i	0.326655i	−0.986572	+	0.163328i	$0.947777\pi$
$978$	0	0
$979$	861246.	0.898591
$980$	0	0
$981$	0	0
$982$	−651141.	−0.675231
$983$	− 690162.i	− 0.714240i	−0.934059	−	0.357120i	$-0.883759\pi$
$983$	− 690162.i	− 0.714240i	0.934059	−	0.357120i	$-0.116241\pi$
$984$	0	0
$985$	0	0
$986$	− 842076.i	− 0.866159i
$987$	0	0
$988$	642656.	0.658362
$989$	− 294600.i	− 0.301190i
$990$	0	0
$991$	981875.	0.999790	0.499895	−	0.866086i	$-0.333372\pi$
$991$	981875.	0.999790	0.499895	+	0.866086i	$0.333372\pi$
$992$	195741.i	0.198911i
$993$	0	0
$994$	−103626.	−0.104881
$995$	0	0
$996$	0	0
$997$	−241946.	−0.243404	−0.121702	−	0.992567i	$-0.538835\pi$
$997$	−241946.	−0.243404	−0.121702	+	0.992567i	$0.538835\pi$
$998$	1.39443e6i	1.40002i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.5.c.h.26.2		2
3.2	odd	2	inner	675.5.c.h.26.1		2
5.2	odd	4		675.5.d.a.674.2		2
5.3	odd	4		675.5.d.d.674.1		2
5.4	even	2		27.5.b.c.26.1	✓	2
15.2	even	4		675.5.d.d.674.2		2
15.8	even	4		675.5.d.a.674.1		2
15.14	odd	2		27.5.b.c.26.2	yes	2
20.19	odd	2		432.5.e.e.161.1		2
45.4	even	6		81.5.d.b.26.2		4
45.14	odd	6		81.5.d.b.26.1		4
45.29	odd	6		81.5.d.b.53.2		4
45.34	even	6		81.5.d.b.53.1		4
60.59	even	2		432.5.e.e.161.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.b.c.26.1	✓	2	5.4	even	2
27.5.b.c.26.2	yes	2	15.14	odd	2
81.5.d.b.26.1		4	45.14	odd	6
81.5.d.b.26.2		4	45.4	even	6
81.5.d.b.53.1		4	45.34	even	6
81.5.d.b.53.2		4	45.29	odd	6
432.5.e.e.161.1		2	20.19	odd	2
432.5.e.e.161.2		2	60.59	even	2
675.5.c.h.26.1		2	3.2	odd	2	inner
675.5.c.h.26.2		2	1.1	even	1	trivial
675.5.d.a.674.1		2	15.8	even	4
675.5.d.a.674.2		2	5.2	odd	4
675.5.d.d.674.1		2	5.3	odd	4
675.5.d.d.674.2		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	675.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{2} +7.00000 q^{4} +19.0000 q^{7} +69.0000i q^{8} +123.000i q^{11} -302.000 q^{13} +57.0000i q^{14} -95.0000 q^{16} -414.000i q^{17} -304.000 q^{19} -369.000 q^{22} -300.000i q^{23} -906.000i q^{26} +133.000 q^{28} -678.000i q^{29} +239.000 q^{31} +819.000i q^{32} +1242.00 q^{34} -740.000 q^{37} -912.000i q^{38} +228.000i q^{41} +982.000 q^{43} +861.000i q^{44} +900.000 q^{46} -2166.00i q^{47} -2040.00 q^{49} -2114.00 q^{52} +1593.00i q^{53} +1311.00i q^{56} +2034.00 q^{58} -2922.00i q^{59} -316.000 q^{61} +717.000i q^{62} -3977.00 q^{64} -4622.00 q^{67} -2898.00i q^{68} +1818.00i q^{71} +3031.00 q^{73} -2220.00i q^{74} -2128.00 q^{76} +2337.00i q^{77} -10450.0 q^{79} -684.000 q^{82} -12633.0i q^{83} +2946.00i q^{86} -8487.00 q^{88} -7002.00i q^{89} -5738.00 q^{91} -2100.00i q^{92} +6498.00 q^{94} +6517.00 q^{97} -6120.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} + 38 q^{7} - 604 q^{13} - 190 q^{16} - 608 q^{19} - 738 q^{22} + 266 q^{28} + 478 q^{31} + 2484 q^{34} - 1480 q^{37} + 1964 q^{43} + 1800 q^{46} - 4080 q^{49} - 4228 q^{52} + 4068 q^{58}+ \cdots + 13034 q^{97}+O(q^{100}) \) 2 * q + 14 * q^4 + 38 * q^7 - 604 * q^13 - 190 * q^16 - 608 * q^19 - 738 * q^22 + 266 * q^28 + 478 * q^31 + 2484 * q^34 - 1480 * q^37 + 1964 * q^43 + 1800 * q^46 - 4080 * q^49 - 4228 * q^52 + 4068 * q^58 - 632 * q^61 - 7954 * q^64 - 9244 * q^67 + 6062 * q^73 - 4256 * q^76 - 20900 * q^79 - 1368 * q^82 - 16974 * q^88 - 11476 * q^91 + 12996 * q^94 + 13034 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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